Abstract
The Sylvester–Kac matrix, also known as Clement matrix, has many extensions and applications. The evaluation of determinant and spectra of many of its generalizations sometimes are hard to compute. Recently, E. Kılıç and T. Arikan proposed an extension the Sylvester–Kac matrix, where the main diagonal is a 2-periodic sequence. They found its determinant using a spectral technique. In this short note, we provide a simple proof for that result by calculating directly the determinant.
References
[1] J. J. Sylvester, Théorème sur les déterminants de M. Sylvester, Nouvelles Ann. Math. 13 (1854), 305.Search in Google Scholar
[2] O. Taussky, J. Todd, Another look at a matrix of Mark Kac, Linear Algebra Appl. 150 (1991), 341–360.10.1016/0024-3795(91)90179-ZSearch in Google Scholar
[3] R. Askey, Evaluation of Sylvester type determinants using orthogonal polynomials, in: H. G. W. Begehr et al. (eds.): Advances in Analysis, pp. 1–16, World Scientific, Hackensack, NJ, 2005.10.1142/9789812701732_0001Search in Google Scholar
[4] W. Chu, X. Wang, Eigenvectors of tridiagonal matrices of Sylvester type, Calcolo. 45 (2008), 217–233.10.1007/s10092-008-0153-4Search in Google Scholar
[5] P. A. Clement, A class of triple-diagonal matrices for test purposes, SIAM Rev. 1 (1959), 50–52.10.1137/1001006Search in Google Scholar
[6] A. Edelman, E. Kostlan, The road from Kac’s matrix to Kac’s random polynomials, in: J. Lewis (ed.), Proc. of the Fifth SIAM Conf. on Applied Linear Algebra, SIAM, Philadelphia, pp. 503–507, 1994.Search in Google Scholar
[7] O. Holtz, Evaluation of Sylvester type determinants using block-triangularization, in: H. G. W. Begehr et al. (eds.): Advances in Analysis, pp. 395–405, World Scientific, Hackensack, NJ, 2005.10.1142/9789812701732_0036Search in Google Scholar
[8] P. Rózsa, Remarks on the spectral decomposition of a stochastic matrix, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 7 (1957), 199–206.Search in Google Scholar
[9] I. Vincze, Über das Ehrenfestsche Modell der Wärmeübertragung, Archi. Math. XV (1964), 394–400.10.1007/BF01589220Search in Google Scholar
[10] W. Chu, Fibonacci polynomials and Sylvester determinant of tridiagonal matrix, Appl. Math. Comput. 216 (2010), 1018–1023.10.1016/j.amc.2010.01.089Search in Google Scholar
[11] E. Kılıç, Sylvester-tridiagonal matrix with alternating main diagonal entries and its spectra, Inter. J. Nonlinear Sci. Num. Simul. 14 (2013), 261–266.10.1515/ijnsns-2011-0068Search in Google Scholar
[12] E. Kılıç, T. Arikan, Evaluation of spectrum of 2-periodic tridiagonal-Sylvester matrix, Turk. J. Math. 40 (2016), 80–89.10.3906/mat-1503-46Search in Google Scholar
[13] R. Oste, J. Van den Jeugt, Tridiagonal test matrices for eigenvalue computations: Two-parameter extensions of the clement matrix, J. Comput. Appl. Math. 314 (2017), 30–39.10.1016/j.cam.2016.10.019Search in Google Scholar
[14] T. Boros, P. Rózsa, An explicit formula for singular values of the Sylvester-Kac matrix, Linear Algebra Appl. 421 (2007), 407–416.10.1016/j.laa.2006.10.008Search in Google Scholar
[15] C. M. da Fonseca, D. A. Mazilu, I. Mazilu, H. T. Williams, The eigenpairs of a Sylvester–Kac type matrix associated with a simple model for one-dimensional deposition and evaporation, Appl. Math. Lett. 26 (2013), 1206–1211.10.1016/j.aml.2013.06.006Search in Google Scholar
[16] Kh. D. Ikramov, On a remarkable property of a matrix of Mark Kac, Math. Notes. 72 (2002), 325–330.10.1023/A:1020543219652Search in Google Scholar
[17] M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly. 54 (1947), 369–391.10.1080/00029890.1947.11990189Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
